To summarize:

The arithmetic number wasn't made up out of thin air. The unit (and arithmetic number) were wrong in the given context (energy).

I claim that the latter is a common tell of a lack of understanding.

You claim that I don't know what I'm talking about since the number wasn't made up.

No, I claim that you don't know what you're talking about because you think the units were wrong in the given context and that the arithmetic values were wrong, which they weren't.

"energy produced from solar systems in 2019 was thought to be about 500 megawatts"

You keep repeating that as if you think it's going to convince somebody, but, in fact, when you divide an amount of energy produced, such as 4 terawatt hours, by a length of time such as a year, you get a power, such as 500 megawatts, which does indeed tell you how much energy was produced over that length of time.

Had it said “500MW-years” it would have been accurate (well maybe, I don’t know). Weird as we normally use MW-hours for energy, but accurate.

They said “500MW”

Is that peak output? Nameplate capacity? 500MWh?

Who knows.

We normally use joules for energy, though sometimes people do unfortunately use cursed units such as foot-pounds, calories, BTUs, megawatt-hours, electron volts, and so on. My analogy of "the distance covered by the runner was 12 miles per hour" is quite precise here. Saying that we're talking about the total energy produced in 2019 clearly excludes your hypothetical interpretations of peak output or nameplate capacity, neither of which would tell you the energy produced in a year.

It's true that someone who didn't understand the area might say "500 megawatts" when they meant "500 megawatt-hours", but the reporter in this case didn't make that error, and anyway 500 megawatt-hours in a year would be 57 kilowatts, which is an implausibly small amount of solar power to be produced by an entire medium-income country. That's more like a single large commercial building.

Domestically people deal with kWh, not joules. That seems a far more useful unit to convey information

Domestically the most common unit of energy is calories in the countries I'm familiar with, but households also frequently measure energy in gallons of gasoline, liters of gasoline, British Thermal Units, "therms", cubic feet of gas, cubic meters of gas, kilowatt hours, cords of wood, bushels of corn, joules, and "tons", meaning tons of ice (often abused as a measure of power, the implicit denominator being "per day"). Additional common units of energy rarely used in households include barrels of oil, "megatons" (this time of TNT rather than ice), toes (ton of oil equivalents), quads (quadrillion BTU), foot-pounds, and, implicitly, horsepower seconds.

This is like the seventeenth century situation where cloth was sold by the "ell", but each kind of cloth was measured by a different ell. The Flemish ell was a different length from the English ell or the Scottish ell. This nonsense survives today in the custom current in the less advanced countries of weighing gold and silver in "troy ounces" and other commodities in a different "ounce" that is about 10% smaller.

The insight underlying the SI system of units is that calculations become much easier if you use a consistent set of units derived in a simple way from a minimal set of base units such as the kilogram and meter. How much kinetic energy is going to kill a 100-kg man falling from a building at 10 meters per second? Trivially, 50 kilojoules. How much energy does a 10-watt lightbulb consume in 5000 seconds? 50 kilojoules. If you are paying US$0.02 per megajoule for gasoline, US$0.01 per megajoule for natural gas, and US$0.04 per megajoule for electrical energy, what's the cheapest way to heat your house? Gas, unless your electric heat pump has a coefficient of performance over 4.

By contrast, if the different prices are expressed in different units, it adds difficulty to every kind of reasoning, to the point that there seems to be a popular misconception that megawatts are not even the right unit for measuring the solar energy production of a country.

Kilowatt hours are an especially silly unit. A watt is by definition a joule per second, so measuring energy in kilowatt hours is similar to measuring distance in mph-minutes. An mph-minute works out to be exactly 88 feet, and a kilowatt hour is exactly 3.6 megajoules. Just use megajoules!